López Jorge M. - Elementary analysis: the theory of calculus

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Kenneth A. Ross Undergraduate Texts in Mathematics Elementary Analysis 2nd ed. 2013 The Theory of Calculus 10.1007/978-1-4614-6271-2_1 Springer Science+Business Media New York 2013

1. Introduction

Kenneth A. Ross 1

(1)

Department of Mathematics, University of Oregon, Eugene, OR, USA

Abstract

The underlying space for all the analysis in this book is the set of real numbers. In this chapter we set down some basic properties of this set. These properties will serve as our axioms in the sense that it is possible to derive all the properties of the real numbers using only these axioms. However, we will avoid getting bogged down in this endeavor. Some readers may wish to refer to the appendix on set notation.

The underlying space for all the analysis in this book is the set of real numbers. In this chapter we set down some basic properties of this set. These properties will serve as our axioms in the sense that it is possible to derive all the properties of the real numbers using only these axioms. However, we will avoid getting bogged down in this endeavor. Some readers may wish to refer to the appendix on set notation.

1 The Set of Natural Numbers

We denote the set {1,2,3,} of all positive integers by . Each positive integer n has a successor, namely n +1. Thus the successor of 2 is 3, and 37 is the successor of 36. You will probably agree that the following properties of are obvious; at least the first four are.

N1.

1 belongs to .

N2.

If n belongs to , then its successor n +1 belongs to .

N3.

1 is not the successor of any element in .

N4.

If n and m in have the same successor, then n = m .

N5.

A subset of which contains 1, and which contains n +1 whenever it contains n , must equal .

Properties N1 through N5 are known as the Peano Axioms or Peano Postulates . It turns out most familiar properties of can be proved based on these five axioms; see [].

Lets focus our attention on axiom N5, the one axiom that may not be obvious. Here is what the axiom is saying. Consider a subset S of as described in N5. Then 1 belongs to S . Since S contains n +1 whenever it contains n , it follows that S contains . Again, since S contains n +1 whenever it contains n , it follows that S contains . Once again, since S contains n +1 whenever it contains n , it follows that S contains . We could continue this monotonous line of reasoning to conclude S contains any number in . Thus it seems reasonable to conclude . It is this reasonable conclusion that is asserted by axiom N5.

Here is another way to view axiom N5. Assume axiom N5 is false. Then contains a set S such that

(i)

1 S ,

(ii)

If n S , then n +1 S ,

and yet . Consider the smallest member of the set , call it n 0. Since (i) holds, it is clear n 01. So n 0 is a successor to some number in , namely n 01. We have n 01 S since n 0 is the smallest member of . By (ii), the successor of n 01, namely n 0, is also in S , which is a contradiction. This discussion may be plausible, but we emphasize that we have not proved axiom N5 using the successor notion and axioms N1 through N4, because we implicitly used two unproven facts. We assumed every nonempty subset of contains a least element and we assumed that if n 01 then n 0 is the successor to some number in .

Axiom N5 is the basis of mathematical induction. Let be a list of statements or propositions that may or may not be true. The principle of mathematical induction asserts all the statements are true provided

(I1 ) P 1 is true,

(I2 ) P n +1 is true whenever P n is true.

We will refer to (I1), i.e., the fact that P 1 is true, as the basis for induction and we will refer to (I2) as the induction step. For a sound proof based on mathematical induction, properties (I1) and (I2) must both be verified. In practice, (I1) will be easy to check.

Example 1

Prove for positive integers n .

Solution

Our n th proposition is

Thus P 1 asserts , P 2 asserts , P 37 asserts

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